Method of and apparatus for determining spacial configuration by means of photographs



Mayl4, 1929.v H. COOKE 1.713.498

METHOD OF AND APPARATUS FOR DETERMINING SPACIAL CONFIGURATION BY mmns OF PHOTOGRAPHS Filed Nov. 24, 1919 8 Sheets-Sheet l INVENTOR May 14, 1929. cool-(E 1,713,498

METHOD OF AND APPARATUS FOR DETERMINING SPAGIAL CONFIGURATION BY MEANS OF PHOTOGRAPHS Filed Nov. 24, 1919 8 Sheets-Sheet 2 INVENTOR RKGTAQ Z 3 BY A TTORA/EY May 14, 1929. H. L. COOKE 1.713.498

METHOD OF AND APPARATUS FOR DETERMINING SPACIAL CONFIGURATION BY MEANS OF PHOTOGRAPHS Filed Nov. 24, 1919 ,8 Sheets-Sheet 3' g f7} 17 f" 5' W C X X 1 V V d I 6' r /zy/l- /5 /g 79 INVENTOR ATTORNEY HXGUA May 14, 1929. COQKE 1,713,498

, METHOD OF AND APPARATUS FOR DETERMINING SPAGIAL CONFIGURATION BY MEANS OF PHOTOGRAPHS Filed Nov. 24, 1919 8 Sheets-Sheet 4 INVENTOR A TTORNEY May 14, 1929. COOKE 1,713.498

METHOD OF AND APPARATUS FOR DETERMINING SPAGIAL CONFIGURATION BY MEANS OF PHOTOGRAPHS Filed Nov. 24, 1919 8 Sheets-Sheet 5 (I mmvnm' flof M BY J r A TTORN E Y -May 14, 1929. coo 1.713.498

METHOD OF AND APPARATUS FOR DETERMINING SPACIAL CONFIGURATION BY MEANS OF PHOTOGRAPHS 8 Sheets-Sheet '6 Filed Nov. 24, 1919 4; F7 Id .1. /A Joirlllampmlmr llPIIIJlJl'l/Illlll IN VEN TOR TTOR/VEV May 14, 1929. 4 Q 1,713,498

METHOD OF AND APPARATUS FOR DETERMINING SPACIAL CONFIGURATIONBY MEANS OF PHOTOGRAPHS Filed Nov. 24, 1919 8 Sheets-Sheet 7 4 I'Z 75 J M IN VEN TOR May 14, 1929. H 1 COQKE -l,713.49 8

METHOD OF AND APPARATUS FOR DETERMINING SPACIAL CONFIGURATION BY MEANS OF PHOTOGRAPHS Filed Nov. 24, 1919 8 Sheets-Sheet 8 i 2 lNl/EN TOR 6 z y Ahab/ Er Patented May 14,1929.

UNITED STATES: PATENT OFFICE.

HEBEWARD LESTER COOKE, or-rnmcmon, new Jami, Assmnoa 'ro Aano sonvxy coma, A CORPORATION or DELAWARE METHOD OF AND APPARATUS FOR DETERMINING SPACIAL CONFIGURATION BY MEANS OI PHOTOGRAPHS.

Application filed November 24, 1919. Serial No. 840,231.

This invention relates to the determination of the spacial configuration or position in space of objects by means of photographs and makes use of the fact that an object when photographed from different points will appear differently placed with respect to other objects (shown in the photographs); In practicing the invention photographs taken from different points are employed in pairs, each photograph of the pair showing the object or objects to be located or region to be mapped, and also showing certain points, the exact locations of which in space are known or may be ascertained in any known way. In taking the photographs the exact relative position of the sensitized surface and the camera lens must be known but the position of the camera in space need not be determined, as this can be ascertained from the known pointsshown in the photographs as hereinafter explained. It is not necessary that the photographs of a pair be taken from the same or similar distances from the objects photographed. By means of the procedures and calculations herein disclosed the exact location of any point or points common to any pair of photographs may be determined from the photographs.

This invention may be employed in any case in which the spacial configuration of objects is to be determined by means of photographs, and can be applied to photographs taken from the ground as well as to those taken from aircraft. The invention is especially useful,

' however, in preparing maps from data supplied by photographs taken from aircraft in flight, and in the description-of the principles and operation of the invention'it will be assumed that photographs taken from aircraft are being dealt with. The application of the method to other photographs will then be considered. As applied to map making, the method is best adapted to plane surveying and the curvature of the earths surface is therefore not considered in the following specification. .The method is applicable to both vertical and oblique photographs and may be em-.

ployed in mapping both mountainous and flat country. I a

In the accompanying drawings which form a part of this specification Figure 1 is a perspective diagram, showing positions of the camera pmhole or lens from which two photographs may be supposed to be taken, the corresponding positions of the camera plates, and the positions of four obects in nature which are included in both photographs Figure 2 is a similar dia ram of certain arrangements to be efi'ecte in the drafting room when the photographs are being employed for the purpose of preparing the map;

F1gure 3 is a diagrammatic perspective view of one form of apparatus which may be utilized in carrying out the invention;

Figure 4 is a longitudinal sectional view of the projection camera shown in Figure 3;

Figures 5 to 9 inclusive are geometrical construction diagrams showing how the equations used in calculating the data for practicing the invention in accordance with the preferred methods are derived;

Figure 10 is a diagrammatic side elevation showing a modified form of adjustable supportlng means for the projection camera, which may be utilized in practicing the in vention;

Figure 11 is a front View of the same;

Figure 12 is a transverse vertical sectional view taken on line XIIXII of Figure 10;

Figure 13 is a longitudinal sectional view through the front part of the camera, the lens mount and the supporting means therefor;

Figure 14 is a fragmentary plan view showing the adjusting means for supporting the rear of the camera;

Figures 15 and 16 are respectively an end view and a transverse section showing an adjustable scale-plate and means for supporting the same in the focal plane of a projection camera;

Figure 17 is a diagrammatic side elevation of a modified form of apparatus which may be used in practicing the invention Figure 18 is a plan view of the apparatus shown in Figure 17, parts being omitted;

Figure 19 is a vertical sectional view of the screen and its supports, forminga part of the apparatus shown in Figures 17 and 18;

Figure 20 is a diagram illustrating the use of the adjustable scale plate shown in Figures 15 and 16, when employed in connection with the apparatus shown in Figures 21 and'22;

Figure 21 is a diagrammatic vertical sectional view of a projection apparatus which may be used for the rapid determination of present invention.

The geometrical principles involved in the invention ma be most easily understood bj; considering allypothetical procedure throug which'the photographs may be supposed to be taken to obtain the necessary information for the preparation of maps. After the general principles have been explained in this way, certain practical methods of applying these principles will be described.

'lhrouglxout the description of this hypo thetical procedure it will be assumed that the photographs are taken and projected by means of a pinhole instead of a lens, and that all the camera projections concerned in this hypothetical procedure are true central (conical) projections. By means of these assumptions the explanation of the hypotlieiiizitzitl procedure will be considerably S1111- 1 e P In this specification the terms project and projection will be taken to refer to orthogonal projections, unless the contrary is specially indicated. Central (conical) projections will be specifically referred to as such.

In Figure 1, let P, and P, represent the two positions of the camera inhole, from which photographshave been t-a en, G, and G, the corresponding position of the plates, and let A B C and W be the positions of four objects in nature which are included in both photographs. The points P, and P, may be designated points of sight or centers of projection. Let R, R, R, R, represent a horizontal plane of reference, say sea level, and let R, R, S, S, represent the plane determined by and containing the three points A B C of which the map positions A B C and heights A A, B B, C C above sea level may be regarded as known. Given the foregoing data concerning points A B C the essential problem of mapping by air pl1otography is to determine the map position W, I

and height W W above the plane of reference of an object such as W which does not necessarily lie in the same plane as the points A B and C.

Let A, B, C, and W, represent the photographic images of the points A B C and W on the plate and A, B, G, and W, the corresponding photographic images on the ratio of P,

paste G,. Produce the rays A,A B,B 0,0

,W to meet the horizontal plane of reference R, B, R, R, in the points A, B, C, W, and similarly produce the rays A,A B,B C,C W,W to meet this lane in the oints A, B, C, and W,'. mm P, and let fall perpendiculars on the plane of reference, meeting this. plane in the points P, and P, which may be desi ated centers of vision or station points Join the point P, to the points A, B, C, and W,, and the int P, to the oints A, B, C, and W,'. ince the line line P, W, on the plane of reference, and the line P, W, is the corres onding projection of the line P W, it f0 lows that the intersection W of the lines P, W, and P, W, is the projection of the int W (the intersection of the lines P, W, and P, W,) on the plane of reference, i. e., W is the map position of W. It is also evident that the eight W W (the height of W from the plane of reference) ma be found by measuring the distance W and multiplying this by the tangent of the angle P,W,'P, (the ratio of the length P, P, to the length P, W,) or by multiplying the length w w, by the tangent of the an 1e P,W,P, (the It appears from the foregoing that if a method can be found of determining for any pair of photographs the positions corrending to P, and P,, P, and P,, and of W, is the projection of thes o taining central projections of these photographs corresponding to the roportions having the configuration A, B, W, and A, B, C, W,, the map position and height of any object such as W included in both photographs may be determined.

To proceed with the description of the hypothetical method of employing the photographs, let Figure 2 represent a perspective diagram of certain arrangements to be effected in the drafting room when the photographs are being employed for the purpose of preparin the map.

On a w ite fiat screen 1', 1', S, S, a triangle .a be is plotted which is an exact reproduction of the triangle A B C, Figure 1, in its own plane R, R, S, S, reduced to some convenient scale, say l/N times the scale of nature. This screen 1', r, S, S, is mounted above the white flat horizontal surface of a table 1', r, 1', r, in such a manner that the vertical distances of the plotted points a b 0 above the surface 1', r, r, r, correspond exactly to the scale l/N with the vertical distances of the points in nature, A B C above the plane of reference R, R, R, B, When this adjustment has been effected the positions of the points a b a form an exact reproduction 4 4 in space, to the scale l/N, of the spacial conhole 72, occupyin the same position'relative e time the photograph was to the plate as at t taken. This roj ection camera is then mounted on an ad ustable stand so that it may be moved and oriented into an position and rigidly retained thereby suita le mechanism.

The light in the pro ection camera is now turned on, and the camera is moved about until the projected images of the photographic images A B (L cast on the screen r 1', S S coincide with the plotted positions a b c a correspondence between the projected images of A B C and a b a similar to that indicated in Figure 1 being of course efiected. Such a fit between the images and the plotted points will be attained when the pinhole position 17, relative to-the plotted triangle a b c is similar to the position of the pinhole P Figure 1, relative to the points A B C in nature. This is evident from the fact that the pinhole and plate in the projection camera occupy the same relative position as the pinhole and plate in the camera with which the photograph was taken, and hence the tribedral angle formed by the emergent rays p a, p 6, p 0 as edges Figure 2 must be identical with the trihedral angle formed by the incident rays A P B P C P as edges Figure 1. It follows then, since the triangle a b c is similar to the triangle A B C that the oblique tetrahedron a b c p, is similar to the oblique tetrahedron A B C P so that the position of 1), relative to the triangle a b 0 is similar to the position of P relative to the triangle A B C. (The scale of the arrangement in Figure 2 below the points 1), and p, is of course 1 /N times the scale of nature shown in the corresponding part of Figure 1.) When this adjustment of the projection camera has been effected, the camera is rigidly clamped in position and retained there during the remainder of the operations. 1

It is to be noted that there are in general several positions of the pinhole p, from which the projected images of the points A B C may be made to coincide with the plotted triangle a b c with the proper correspondence between points. There is therefore an apparent ambiguity in the method described of locating the point P It has been found by actual experiment however that this ambiguity will occasion no serious trouble in locating the proper position of. P provided the conical angle subtended by the three points A B C, Figure 1, viewed from the point P is reasonably large, and provided that the point P Figure 1, does not lie in the immediate neighborhood of any of the three perpendiculars erected on the points A B C normal to the plane R R S S A very little practice in the use of this method will enable the operator to detect at a glance when the projection is made from a wrong position of 1),, as the persective of the images projected from any of the wrong positions is distorted in an absurd manner which is apparent from. a very' casual inspection. The plate G, is now mounted in a second projection camera similar to the one employed with the p ate G and with the pinole p, in the pro er position relative to the plate G, as s eci ed in the case of the pinole p and p ate G,. This second projection camera is then moved around until the proper position of the inhole 12 is arrived at, as indicated byxa coincidence of the projected images of t e photographic images A 13, G with the plotted points a b c on the screen 1', 1', S S,, the difliculty in the ambiguity of theposition of p, scribed above. This second projection camera is then clamped rigidly in position, and maintained thus during the remainder of the operations. he screen 1', 12,. S S is now removed, the two projection cameras being left in position. The 1projection camera on the left, Figure 2 is t en turned on, and the images of the photographis images A B C W centrally projected on the horizontal surface of the table 1', 1', r, 1', will then occupy the positions a, b, 0, corresponding with the configuration of the points A, B, C, Figbeing overcome as de-' urel, on the reduced scale l/N times that of nature. The projected image of the object W, Figure 1, will be found at the point w corresponding on the scale l/N with the position W Figure 1. The positions of these points a, b, a, w, are marked, the position of 17 the rojection of the pinhole position 1), on the p ane of the table located by means of a plumb bob let fall from 72,, and height p, 1),- measured. The points a, b, e,

w," are then oined to the point p, by straight lines, drawn on the table.

The light in the projection camera on the left is now turned oil, the light in the projection camera on the right turned on, and the positions of the points a, b c, w, Figure 2, corresponding to the points A, B C, W, Figure 1, the position of the point 7),, the projection of the point p, on the plane of the table, and the height 17 7), determined in a manner similar to that of determining the points a, b, 0, p and p, as already described. Straight lines are then drawn from p, to a, b, a, and w,.

The intersections of the lines 12 a, and p, a, p, b, and p, b, p, c, and p, a, will then determine the ma positions A B C of the points A B C Figure 1, to the scale of l/N. The map position, to the same scale, of W will evidently be at the intersection w of the lines 12 w, and p, 20 or these lines produced when W is below the plane of ref-' erence and the height W W, to the same scale, is found by multiplying the measured dis? tance w, w, by the tangent of the angle 12;

w, p, or by multiplying the measured distance w w, by the tangent otthe angle p,

w, 1),. Thus the position in space of the object W relative to the points A B and C Figure 1 is completel fixed. The positions in space of all other 0 jects which appear in both hotogi'aphs may be similarly located, and t e region around the triangle A B mapped b this method.

It is to noted that if the image of an object in the plane-of-reference projection lies between the map position of the object and the centre of vision (station point) of the projection, the object lies below the plane of reference. Its distance from the plane of reference is calculated as before by multiplying the distance of the image from the map position by the height of the point of sight from the plane of reference, and dividing by the distance of the image from the centre of vision.

Although it is theoreticall possible to carry out the foregoing proce ure, in actual practice the difiiculties would be found to be very great, as it would be a very slow and tedious process to adjust the two projection cameras accurately to the correct positions as shown in Figure 2. Special apparatus has therefore been designed for carrying out a process equivalent to the hypothetical pro cedure described, in which the adjustments ma be accomplished with ease and certainty. A escription of one form of this apparatus will now be given.

it is now assumed that the photographs to be dealt with are taken and projected with lenses, and not with pinholes.

The apparatus is shown in perspective in Figure 3. There are three main parts, the projection camera and mounting 1, shown on the left, the rojection screen and mounting 2, on the rig 1t, and two horizontal parallel rails 3, shown in the lower part of the figure,

upon which two carriages supporting the that employe projection camera and screen may be moved backwards and forwards parallel to the rails 3.

The lens of the projection camera is placed at 4, the late holder at 5, the condensing lens is wit in the body of the camera behind the plate holder at 6, and the source of illumination is supported centrally on the main camera axis 4-7 by the mount 8. These four parts are all carried by the rigid framework of the camera.

The lens 4, referably of the same type as in the aircraft in taking the photographs, is mounted with its axis in a position relative to the plate G identical with the position of the line of the axis of the photographic lens-relative to the plate G, at the time the plate G, was exposed, and in the usual case coinciding with the axis 4-7. It is carried in a focusing screw mount 9 (Figure 4) so that the distance between the lens and the plate carried in the plate holder may be adjusted by moving the plate towards and from the lens. The plate holder 5 is in the form of a circular ring and is mounted in the framework of the camera in such a manner that it may be rotated about the camera axis i -7, the plane of the plate remaining normal to this axis during this rotation. As shown in Figure 4 the rear part of the lens mount is externally threaded and slidingly fitted upon the tubular portion 10 at the front of the camera. An internally threaded adjusting ring 11 is mounted to rotate freely on the tubular portion 10 and has a screw-threaded engagement with the lens mount 9. By rotating the ring 11 the entire camera and with it the plate holder and plate may be adjusted with respect to the lens mount whichis supported by the gimbals described below. The condensing lens is rigidly mounted within the body of the camera, with its axis coinciding with the axis 47. The mount 8 for the source of illumination is adjustable, so that the source of illumination may be moved backwards and forwards along the axis 47,-for the purpose of securing even illumination of the projected image.

The projection camera is supported on a carriage 15 fitted with wheels 16 which enable it to be shifted backwards and forwards along the rails 3. A rigid metal framework 17 clamped to the carriage 15, terminates at its upper extremity in a square framework 18. Two screw gimbal pins 19, pass through this square framework and terminate in conical hearings in the gimbal ring 20. Two screw gimbal pins 21 (the far one not shown) pass through the gimbal ring 20 and terminate in conical bearings in the mount 9 of the lens 4. The axes of the two gimbal bearings coincide in the axis of the lens 4 in the position of the external principal point of the lens, the female portion of the bearings of the gimbal pins 21 being drilled in the lens mount in such a position as to secure this adjustment.

It is shown in treatises on geometrical optics that there are two points, called principal points, on the axis of a lens, such that if a straight line be drawn from an object on one side of the lens to one of these principal points, and a second line be drawn, parallel to the first line, through the second principal point, this second line will pass through the position of the image of the object from which the first line was drawn. In this specification the principal point to which rays external to the camera are drawn in this method of locating the image will be called the external principal point, the other principal point being called the internal principal point. This convention will be adhered to whet-her the camera be a photographic or a projection camera.

The main weight of the projection camera rests on a metal rail 22 bent into the are of a horizontal circle having its centre on the vertical line through the axes of the gimbal ins 19. This bent rail 22 is rigidly attache to the vertical metal rod 23 which may be raised and lowered by a rack and pinion mechanism 0 erated by t e milled head- 24. The screw c amp 25 enables any vertical adjustment of the rod 23 to be rigidly maintained.

As the rod 23 is raised and lowered the projection camera will swing about a horizontal axis coinciding with the axis of the 'mbals 21. This axis will be referred to as t e horizontal axis of the rojection camera. As the camera is slid si eways in either direction along the bent rail 22 it will swing about a vertical axis passing through the gimbals 19. This axis will be referred to as the vertical axis of the projection camera. As the plate holder 5 is rotated the plate and the projected image of the plate will rotate about the main axis 4- 7 of the projection camera,

this axis being normal to the plate and passing through the intersection of the horizontal and vertical axes of the projection camera. As the carriage 15 is slid backwards and forwards along the rails 3 the camerawill experience a motion of pure translation parallel to the rails 3. It'is seen therefore that this mounting of the projection camera enables the axis of the camera to be pointed in any direction within wide limits, the plate and projected image to be oriented in any manner about this direction as an axis, and

the camera to be brought into any position, subject only to the constraint that the external principal point of the lens'remains coincident with a horizontal line passing through the intersection of the vertical and horizontal axes of the projection camera, this line being parallel to the axes of the rails 3. This is equivalent to mounting the camera so that it may be rotated about three independent axes coinciding in a principal point of the lens, and also so that it may experience displacements of pure linear translation parallel to a given line.

Passing now to the screen and mount 2, shown on the right in Figure 3, this consists of a flat surface 26, preferably of glass with a ground glass surface on the side facing the projection camera, so that the image cast by the camera on the screen may be viewed through the glass from the side of the screen remote from the camera. The-screen '26 is held securely in a frame 27 which is supported by trunnions 28 (the far one not shown) resting in bearing grooves in the framework 29. I bearings is arranged so as to be at exactly the same height above the rails 3 as the intersection of the horizontal and vertical axes of the projection camera at 4. This trunnion mounting enables the screen 26 to be rotated about a horizontal axis coinciding with the trunnion axis. This axis will be referred to as the horizontal axis of the screen. The

The axis of these trunnion framework 29 is rigidly clam d to the circular table 30, which is capa le of rotation about a vertical axis by means of a conical bearing 31 passing down centrally through the b0 y of the main support 32. This axis will be referred to as the vertical axis of the screen. This main support 32 is rigidly secured to the carria e 33, which is similar to the carriage 15, an runs on wheels 34on the rails 3. The vertical and horizontalaxes of rotation of the screen mounting are so arranged that they intersect in a point, the line joining the point of intersection to the point of intersection of the vertical and horizontal axes of the camera being parallel to the axes of the rails 3. The line oining these two points of intersectionwill be referred to as the axis of linear motion. The screen 26 is so placed that when its plane is vertical both the vertical and horizontal axes of the screen lie in the plane of the surface of the screen which faces the projection camera. The point of intersection of the axes in the plane of the screen will be called the screen centre. A member 35 having a circular scale engraved thereon is rigidly attached to the frame 27, and a pointer 36 cooperating with this scale is rigid ly attached to the framework 29 so that the angular displacements of the screen 26 about the horizontal axis of the trunnions may be determined by the operator. Similarly, a circular scale on the circular table 30, co-operating with a pointer attached to the main. sup port 32 enables the operator to determine the angular displacements of the screen 26 about the vertical axis. These two pointers and scales are so arranged that when the plane of the screen is normal to the axis of linear motion the reading on each scale is zero.

One of the rails 3 is graduated with a linear scale in some convenient unit, say millimeters, and pointers 37 and 37,, attached to the carriages 15 and 33 respectively, cooperate with this scale. This scale may be arranged with increasing numbers running from left to right (away from the projection camera). A preliminary adjustment is carried out in which the carriage 15 is moved into such a position on the rails 3 that the reading of the pointer 37 attached to the screen carriage 33 on the linear scale on the rail 3 indicates the horizontal distance from the intersection of the vertical and horizontal axes of the camera to the intersection of vertical and horizontalaxes of the screen. 7

A method of employing the projection apparatus shown in Figure 3 will now be de scribed. Suppose that the two photographs shown in Figure 1 have been taken, these two photographs including the control triangle A B C of which the positions and heights of thethree points may be regarded as known, and the object W, whose position is to be determined. Let the photographic images of the points A B C on the plate G, exposed from the lens position P, be A, B, (3,, Figure 1 and let A be the highest and B the lowest oint of the-control triangle in nature, C being the point of intermediate height. On the screen 26, Figure 3, a triangle a b 0 is plotted, which is anexact reproduction to the scale l/N previously referred to, of the control triangle A B G in nature in its own plane and is thus an exact reproduction of the triangle 0 b c, Figure 2. The dlmenslons of the triangle a b 0', may be determined from the map 'sitions and elevations of the ints A B and?) in anysuitable manner, for mstance, by determining the lengths of the sides of the triangle b means of equations I given below. The point 0' corresponding to the point of intermediate height C in the control triangle in nature is plotted on the screen 26 at the screen centre and is therefore on the axis of linear motion. One of the other points of the plotted trianglea' b a, sa a, corresponding to the highest point A o the control triangle, is conveniently plotted on the same horizontal line as c, the two points a and cv thuslying on the horizontal axis of the screen 26. The third point I) corresponding with the lowest point of the control triangle in nature is conveniently plotted above the line joining a and c. a

One of the photographs, say G,, Figure 1, is now mounted in te late holder 5, Figure 3, with the emulsion side towards the projection lens 4. This lens is of the same t pe as that used in exposing the plate G,, at

referably of shorter focal length, this focal ength being determined by the fact that the projected image-must be brought to reasonably shar focus on the screen 26 when the relative a justment of screen and camera is such that the gzojected image of the control triangle may made to fit the plotted triangle a' b c, and a reasonably lar e diaphra is used in conjunction with t e lens 4. 'l li: focusing screw mount of the projection lens 4 is now adjusted so that the relative position of the internal principal point of the lens is placed with respect to the plate G, in a manner identical with the corresponding arrangement of lens and plate at the time this plate was exposed.

The source of illumination in the projection camera is now turned on, and adjusted to a position givin even illumination of the image thrown on t e screen. This image is examined, and the plate holder 5 rotated until the linejoining the projected images of the photograpgic images A, and C, Figure 1, which are to fitted to the plotted points a and c respectivelfy is horizontal with the ggojected image 0 B Figure 1, which is to fitted to the plotted point 6', above the line 'oining the images of A, and G,. The milled ead 24 is now operated, shifting the supporting rod 23 vertically until the line joining the images of A, and C, projected on the screen 26 coincides with the line joining a. and c, on the screen. The projectioncamera is now slid alon the, rail 22 until the projected'image of Figure 1, coincides with the plotted position of a at'the center of the screen. This is the final adjustment of the projection camera about its vertical and horizontal axes and the camera is left in this position during the remainder of the operations on the plate G,.

It is obvious from the manner in which the apparatus is constructed and adjusted, that after the foregoin adjustments of the rojection camera have been effected the projected image of the photographic image C, will remain coincident with the lotted point a and the projected image 0? the photographic image A, will remain on the horizontal line passing through the plotted points 85 a and 0' no matter how the screen 26 may be moved about or oriented on its mounting.

The carriage is now moved into such a position that the projected image of the photographic image A,, Fi ure 1 coincides with the plotted oint a. he screen is then rotated about its horizontal axis to ascertain if there is any orientation about this axis which will bring a coincidence between the projected image of 13,, and the plotted point I) 05 while the projected images of A, and C, remain coincident with a and c. If no such orientation is found the screen is rotated through an an le about the vertical axis, the carriage 33, s ifted to secure coincidence 10 again between the image of A, and the plotted oint a, and another attempt made, by rotatmg the screen about the horizontal axis, to secure a simultaneous coincidence, of the projected images of A, B, C, with the plotted 5 points a b c. The trials are repeated in this manner until the proper coincidence has been attained. Experiment has shown that these operations can be eiiected with rapidity and certainty.

When these adjustments have been made the following four uantities are recorded (1) The scale, re ative to nature, to which the triangle a b c is plotted.

(2) The reading of the pointer 37,, on the 115 linear scale graduated on the rail 3, which fixes the distance from the point of intersection of the axis of rotation of the screen mount to the intersection of the axes of the camera.

(3) The reading on the circular scale on the table 30.

(4) The reading on the circular scale on the member 35.

By these four quantities and the coordinates 125 in space of the-three points'of the control triangle A B C it is possible, as will be shown later, to calculate the positions of the points P, and P,, Figure 1.

The coordinates in space of thepoints P, 130

may now be alternately P, A B C Figure 1 being known, it is possible by the usual methods of solid geometry to calculate the positions of the points A, B, 0,, Figure 1.

When this has been done, the triangel A, B, C, to the scale l/N, is plotted on the screen 26 in the same way that the triangle 0 was plotted, C, occupying the position 0, A, being on the same horizontal line at C, and B, occupying the upper position. The screen mounting and carriages are then adjusted so that the projected images of A, B, C, cast on the screen, coincide with this new plotted triangle, the method of carrying out this operation being similar to that described for obtaining a coincidence with the triangle (1 I) c. The projection cast on the screen 26 after this operation has been eflected will be termed the plane-of-reference projection of the plate G,, A positive photographic reproduction of this plane-of-reference projection of the plate G, is now obtained my making an exposure on a photographic plate substituted for the screen 26,

, the adjustments of camera and screen holder remaining unaltered durin the substitution of the photographic plate tbr the screen.

The late G, 1s now put through a process precise y similar to that employed with the plate G,, in which, by means of a coincidence between the projected images of the photographic images A, B, 0,, Figure'l, with the plotted triangel a b a, Figure 3, data are supplied for the calculation of the positions of the points P, and P,, Figure 1: and subsequently a photographic positive of the plane-of-reference pro ection of the plate G,, corresponding to the configuration A, B, C," W,, Figure 1, is obtained, by a. process similar. to that employed in obtaining the photographic positive of the plane-.of-reference projection of the plate G,.

A flat surface on which the'map is to be drawn is now prepared, with the points P, P, A, B, G, A, B, C, the scale l/N. Bv normal optical projection or other suitable means the plane-of-reference projections of the plates G, and G cast on or transferred to the map surface without appreciable distortion, the plane-of-reference projection of plate G, being correctly located by'means of the plotted points A, B,, (1,, which also appear on the plane of-reference projection, and the corresponding projection of plate G, being correctly located by means of the plotted points A, B, 0,, in a perfectly obvious manner. The projection of the-plate G, will then show the position of the point W, corresponding with the object W, while the projection of the plate G, will show the corresponding position W,. The intersection of the lines P, W, and P, W, will then give the map position of W. while the measured distance W W,, multiplied by the height P,

plotted on it to.

P, divided by the measured distance P, W,, or the measured distance W W, multiplied by the height P, P, divided by the measured distance P, W, will be equa to the height W W of the object W above the plane of reterence, these measurements of course being on a scale 1 /N times that of nature. The position in space of WV is thus determined. All objects such as W appearing in both photographs may be similarly located, and in this way data obtained for the preparation of a map of the region common to the two hoto a hs G G p sho i ld the bpeiator be in doubt as to whether the two plane-of-reference projections have been obtained-from the correct centers of projection, owing to the ambiguity of these positions when onl three control points are used, he can satisfy liimself on this point by determining the height of an object such as W by means of both displacements W, W, and'W as hereinbefore described. If these two determinations of the height of, \V agree, this is a proof that the correct centers of projection have been chosen for the two plane-of-reference projections, while a lack of agreement, beyond the range of experimental error, will indicate that at least oneplane-of-reference projection has been obtained from the wrong projection center. The object chosen for this check on the correctness of the centers of projectionshould not lie near any of the three lines inter-connecting the three control points in nature. While actual experiment has shown that this ambiguity in the position of the correct-cc ter of projection in the three pointv method of control will occasion no difliculty if the precautions hereinbefore mentioned are observed, this method of checking the correctness of the projections answers any objections which may be raised on this score.

It is not necessary that the, same control triangle should be used in obtaining projections of the'two photographs to be used in conjunction with each other for mapping purposes by the method described. It is only necessary that eachphotograph show three points, the positions of which are known, so that a control triangle may be established for each of the photographs, and that the two photographs include a common area to be mapped. This follows because the primary function of the control triangles is to enable the centers of projection P, and P, and the station points P, P, to be determined, these points corresponding with the positions of the cameras at the time the photographs were taken and obviously being the same irre spective of what particular control triangles may be utitlized to determine them.

The mathematical theory of the method will now be dealt with. The discussion will begin with an examination of some of the geometrical properties of the control triangles employed in projecting the photographs,

- will see the highest .point on his left and the lowest on his right. In plotting a control triangle in which all the points are at the same height, any of the points may be taken as the high point, and any other point as the low point. If two points in a control triangle are at the same height, either of these two 'points may be taken as the higher.

In Figure 5, which is a perspective diagram, let A B 0 represent the control trian 1e dealt with in the previous discussion, A eing the highest oint called the high point, B the lowest, cal ed the low point, and C the point of intermediate height called the intermediate point (A B C Figure 5 is a positive triangle, since to an observer standing at C, A appears on the right and B on the left). Let OX OY OZ represent the axes of rectan ular coordinates to which the positions of A B C are referred. Project the triangle A B C orthogonally on the horizontal plane passing through C, A projecting to A and B to B, and let G M be the line of intersection of the plane of A B C with the horizontal plane through C. M, which will be called the mid oint of the triangle, is that point on the ine joining the high to the low point which is at the same height as the intermediate point. This line C M will be called the mid line of the triangle, From A let fall a normal on the mid line C M or C M produced, meeting this line in E. Project the figure A B C M E on the plane z=o to 'form the figure A B, G M E Produce the line E A to A" making E A" equal to A. A" 'is then the position which would be occu ied by A if the triangle A' B C were rotate about the line C M- into the horizontal plane, B" being the corresponding position which would be occupied by B. Project the points A and B" on the plane z=o, A," and B," being the positions thus found. Join M to C A to G and A," to C and produce these lines to meet the axis of X in the points U V and V respectively. By the construction of the figure it is seen that the angle A E A, designated 1;, is equal to the dihedral an le between the plane 0 the triangle A B and the horizontal. Let be the angle M O A and X8 the projection of this angle on the horizontal plane. The angle M C A is equal to x and is equal to the angle M, C A The angles which the lines M C A C and A C make with the axis of X are designated can and a respectively, as shown. The angle A C A is designated Let (02 3 5,.) (m; 3 a and (w 3 2 be the rectangular coordinates in space of the points A B and C. It is then evident, by the ordinary methods of solid geometry that the following relationships hold The lengths of the three sides of the triangle are given by III. tan 1.: (yr, yc)/ M o) 'IV. tan as (31.4 31c) .4 a)

V- xa= i (u n) VI. sin 1 (2,4 0) L VII. tan 1) tan a sin x VIII. tan x tan x cos 1 XI. a a x Equations I, II, III, VII, VIII, and IX are employed in the calculation connected with the projections, while IV V, and VI are used only in deriving vii, V311, and IX. In Equation V, the upper of the alternative signs is to be used in connection with positive triangles, the lower in connection with negative triangles. The angle x is taken as the angle included between the mid line and the-line joining the high 'to the intermediate point in the plane of the triangle, whether the triangle-is positive or negative. The signs of the angles x 17 are taken to correspond with the sign of the triangle in which they occur, being positive in positive triangles; and vice versa.

In dealing with the mathematical theory of the projection apparatus shown in Figure 3 it is first necessary to establish a convention as to the method of plotting triangles on the screen 26am] as to the signs of the angles indicated on the two circular scales of the screen mountin A convenient convention for plotting t e triangles, but not the only possible one, will be to plot the intermediate point,- corresponding,to C, Figure 5, at the screen centre, to plot the hi h point corresponding to A, Figure 5, on t e horizontal .axis of the screen to the right of the screen centre as viewed from the projection camera.

The third point, corresponding to the lowest is ploftedabove the iorizontal line oin ng the other two plotted points if the triangle is.

positive one, and below this line, if the tri-.

angle is a negative one. The triangle a b a, Figure 3, representing the positive triangle A B C shown in Figure 1, is plotted in accordance .with this convention.

A suitable convention with regard to the signs of the'angular displacements of the screenmounting will be to reckon angular displacemehts about the horizontal axis positive when the upper part of the screen is advanced toward the projection camera, and to reckon angular displacements about the vertical axis positive when the right hand.

side of the screen, as viewed from the proj ection camera, is advanced-towards the projection camera; and vice versa. The zero position for both angular adjustments of the screen as previously stated is when the plane of the screen is vertical and normal to the axes of linear motion 4-0, Figure 3. In accordance with this convention both angular displacements correspondin with the position of the screen shown in Figure 3 are positive.

The exact nature of the operations carried 1 out with the projection apparatus will now be considered. After the projection camera has been correctly adjusted there are three adjustments to be made to the screen affecting (1) the distance between the camera and screen (2) the orientation of the screen about the vertical axis, and (3) the corresponding orientation about the horizontal axis; these adjustments resulting in an exact coincidence between the projected and plotted triangles. It may. be supposed for convenience that these three operations are carried out completely in the above order, beginning with the plane of the screen vertical and normal to the axis of linear motion. Consideration will show that. with the screen in this initialposition the orthogonal projection of the lens centre 4 on the screen 26 will lie at the screen centre, and this condition will continue to hold after operation (1) has been eli'ected. During operation (2) the orthogonal projection of 4 on the plane of this screen travels along the horizontal axis of the screen, arriving when the operation is completed at some such point as 38, Figure 3. During operation (3) the orthogonal projection 0154 on the plane of the screen travels from this point 38 in a direction at right angles to the horizontal axis of the screen, arriving when the operation is completed at some such point as 39, Figure 3, where 0'-38 and 3839 are at ri ht angles. It is evident that operation (2) is equivalent to maintaining the plane of the screen (and therefore of the plotted control triangle) fixed, and swinging the external principal point of the lens 4 through the arc of a circle having the screen centre the intermediate point of the plotted triang e) as centre, theplane of this circle intersecting the plane of the screen (and therefore of the plotted triangle) in the horizontal axis (the line joining the high to the intermediate point in the plotted triangle), the plane of this circle being normal to the plane of the screen (and therefore normal to the plane of tie plotted triangle). Similarly'operation (3) is equivalent to maintaining the plane of the screen (and therefore of the plotted triangle) fixed, and rotating the lens centre 4 through the arc of a circle lying in a plane normal to the plane of the screen (and therefore normal to the plane of the plotted triangle) and normal to the plane of t e circle considered in the operation equivalent to operation (2) the centre of this second circle lying on the horizontal axis of the screen (and therefore on the line joining the high to the intermediate point of the plotted triangle, or this line produced).

If in these equivalent operations the angles through which the lens centre 4 is swung are equal to the angles through which the screen is, actually rotated in operations and (3), due regard bein paid to the signs of these angles, then the final relative configuration in space of the points 4 a b a resulting from the equivalent operations will be identical with that effected by the actual operations (2) and (3). It will therefore be assumed that these equivalent operations are carried out, as the geometrical representation of these equivalent operations is considerably simpler than t(hz;.t involving the-actual operations (2) and Figure 6 is a perspective diagram, illustrating the principles of the operation of the projection apparatus. Suppose that a correct coincidence has been effected between the projected image of-the control triangle A B C appearing on the photographic plate Gr and the corresponding plotted control triangle a Z) c, by means of the projection apparatus shown in Figure 3 as already described. Let the distance between lens centre and screen centre corresponding to this coincidence be D, the orientation of the screen about the vertical axis be the orientation of the screen about the horizontal axis be 6, and the scale of the plotted triangle be l/N.

Let the triangle A B C Figure 6 represent the control triangle a b c plotted on the screen 26, Figure 3 and let the position of P Figure 6 relative to A B C correspond ,with the position of the external principal point of the lens 4 relative to the plotted triangle a b 0 in the projection apparatus when the correct coincidence between the projected and the plotted triangles'has been effected. Let the position of the axes of coordinates OX OY OZ relative, to the triangle A B G correspond with the position of the Y and the point P, moves to P,,.

- and will be denoted as x erence z==o, the

ing J J P,=0, the angle determined by A B C and let Lbs the dihedral angle included between e plane. of A B C and the horizontal. From P, let fall a normal P, K on the line C M or G M produced. Rotate the system P, A B C through the angle 1; about the axis 0 M, so that the triangle A B C assumes the horizontal position A" B" C, The relative configuration of the four points P, A" B C .is now identical with P, A B C, so that A B" C may now be considered to represent the plotted triangle a b c on the screen 26, and P, the externalprincipal point of the lens 4, the apparatus still-being adjusted for-coincidence of projected and plotted triangles. Project the points P, P, K C A B M orthogonally on the horizontal plane of refpositions then found bein P, P, K C A B," M,. Produce C 0 and C A," to meet the axis of X at U and V respectively. Using the notation emplo ed in connection with Fi ure 5, M X is the angle A, V X is t e angle a, and M (3 A, is the angle Join P, to K. By construction the angle P, K P, is equal to the-dihedral an le 1,. Through K draw the. vertical line K K. The angle P, K K

will be termed the mid-line declination This angle may be defined as the dihedral angle which the plane including the mid line of the control triangle in nature and the aeroplane position makes with the plane which intersects the plane of the control triangle normally in the mid line.

Through C draw the vertical line C H making CH=D, the distance between screen centre and lens centre, determined by 0 eration (1) of the projection a paratus. vi ah C as centre and C H as ra ius describe the are H J of a circle lyin in the vertical plane pizssing through 0 A making the angle C J p the angle throu h which the screen was rotated in operation 2). From J drop a perpendicular on the zero plane which will meet 0,, A," or G A, produced, in J, and C A" or C A produced, in J. With J as centre and J J as radius described the arc J P, of a circle lying in the vertical plane normal to the plane of the angle H C J, and makeration (3) of the projection ap aratus. is the point previousllg located y swinging P, about the centre through the angle 1; in the vertical plane.

It is necessary of course that thesigns of the angles (,0 and o' in the above construction should agree with the corresponding angles of ad'ustment of the screen in operations (2) and 23), using whatever convention of sign has been adopted. Using the conventions which have previously been stated it is seen relative to the contro triangle in nature.

Since in the figures the triangle A B C s placed relative to the axes of coordinates 1n a similar manner to'the arrangement of the control trian le and axes of coordinates in nature, it foll ows that the coordinates of the point P, in the diagram relative .to the axes of coordinates shown will., correspond with the position of the aeroplane :relative to the axes of coordinates in nature. Thus if all the linear dimensions of the dia am are increased in the ratio N: 1, A B may be 1'82 garded as the-control triangle in nature, with P, the aeroplane position, 0 the origin of coordinates and P, the projection P, on the horizontal plane of reference. It will be assumed that this change in the dimensions of the dia am has been effected, the length H 0 being 0 an ed from D to N D. The problem then is to nd the coordinates of P, and P, Figure 6, corresponding to the points P, and P, Figure 1.

If in Fi re 6 H G be taken as, equal to N D, it is o vious from the construction that the following relationships hold sin 4 sin x+cos sin 9 cos x cos d: cos 0 which fixes the value of the mid-line declination. The mid-line declination all is reckoned positive when P, lies to the left of the mid line, looking in the direction of the mid point M as viewed from the intermediate point C.

Remembering that P, P,=N D cos cos a it may beshown by plane trigonometry that the following relationshi s hold in the plane figure P, P, P, K o

The rectangular coordinates su 3 Zp of P,, the aeroplane position, may now be written down, the a: coordinate bein obtained by adding to the a coordinate of the components of the displacements C J o J o P, and P, P, parallel to the axis of a), the coordinate similarly being obtained by a( ding to the y coordinate of the C the. components of these same displacements arallel to the y axis. The 2 coordinate-of I, is given by adding the height P P to the a coordinate of C. We then obtain the expressions XI. 2' x,,=a:,+ND [sin cos a+cos sin 0 sin a+eos cos 0 sec Il [sin (l +31) sin a] sin a] y,,=y,+ ND [sin sin a+60S sin0 cos a+cos 1: cos flsecd/[sin (+y) sin Moos p] These equations determine the aeroplane position, correspondin to the position P Figure 1. The position corresponding to P Figure 1 is of course obtained by putting 2,, in the third equation equal to zero orwhich is simple in application has therefore been devised.

The terms ar 51 and .2, are of known sign, from the data concerning the control triangle. The second term in the expression for 2., will practically always be positive in the case of aeroplane photography. Should the unusual case occur, however, in which it is negative, this will be indicated by the fact that the value of +1 does not lie between the limits +90 and 90 (0. f. Figure 6). It appears therefore that the only difiiculty with regard to sign will occur in connection with the terms within the square brackets in the expressions for x and 3 The signs of these terms may be found as follows.

Consider, as in Figure 6 that the plane of the control triangle has been rotated about the mid line into the horizontal. Plot on squared paper the projection of this rotated triangle on the horizontal plane of reference showing its position relative to the m 1 and y axes of coordinates. Figure 7 shows the triangle A B C so plotted, the lettering of Figure 7 corresponding with Figure 6. Produce the line A C joining the high and the intermediate points of the triangle, and produce M C these two lines produced meeting the w axis in V and U respectively. From O lay off along the line C A or C A produced some convenient length C J this line being drawn from C toward A, if (p is positive, and from G away from A if 1; is negative. From J draw a line J, P of any convenient length, at right angles to C A this line being directed toward the" same side of C A as B if 6 is positive, and on the other side of C A, if 0 is negative. From 1 draw a line P P of any convenient length at right angles to the mid line C M If the control triangle is positive this'line is drawn toward the left of an observer standing at O and facing M and vice versa. The directions of the three lines C J J P and P P taken in order correspond with the displacements determining the values of the three terms within the square brackets in the expression for w,, and y,,, taken in order. It is possible to determine the sign I of the w and 3 components of these three lines by simple inspection. The figure as drawn corresponds with a positive triangle. a negative value of (p and a positive value of 0. By simple inspection the sign of the corresponding terms within the brackets will be for w,, and for y, Any practical example may be dealt with by this simple method, which has the advantage that the function of all angles may be regarded as positive and that a graphical check is pro- 6 videdfor the calculated values of a and X as dependent on The method of determining the central projection of the control triangle in nature'on the horizontal plane of reference from the aeroplane position as centre of projection will now be considered. If P is the centre of projection, A B C the control triangle, and if the height of the plane of reference is 2 the coordinates of the projections A B and C of A B and C are given by the following equations.

XII.

l If the horizontal plane of reference is sea level, a is put equal to zero.

The points A B C determined by equations XII are shown similarly designated in Figure 1.

The triangle formed by the points determined by equation XII is now plotted to the scale l/N on the screen 26 with C, at the centre and A, on the horizontal screen axis. By means of the projection apparatus the control triangle in the projected image of the plate is then made to coincide with this plotted triangle. This having been accomplished,

' the plane-of-reference projection of the plate G, to the scale l/N already referred to.

In a similar manner a photographic positive is obtained of the plane-of-reference projection of the plate Gr to the scale l/N.

To briefly recapitulate the steps to.,be carried out in practicing the method described above in detail, the procedure is as follows 1. Procure two photographs taken from diiferent positions, each photograph showing the area to be mapped and also showing three points, the map positions and elevations of which are known, the three points in each photograph determining a control triangle for that photograph. The exact relationship of the plate to the camera lens at the time each photograph was taken must be known.

2. Calculate the dimensions of the control triangles from the known coordinates of the three known points in accordance with Equations I.

3. Plot the control triangle for one of the photographs on the screen 26, in the manner described, with the point of intermediate height at the screen center and the high point of the triangle on the horizontal axis of the screen. 7

4. Place the first photograph'in a projection apparatus in the same relationship to the lens of such apparatus that the plate occupied with respect to the camera lens at the time of exposure.

5. Project the first photograph centrally on the screen and adjust the camera and screen, as described above, to secure coinci dence between the projected image and the plotted triangle.

6. By reading the scales of the apparatus determine the distance between the lens and the screen center, and the vertical and horizontal deflections of the screen.

7 Calculate the airplane position or center of projection P, and the orthogonal projection P, of this point on the plane of reference by means of Equations II to XI.

8. Calculate the positions of A, B," C,

' representing the central projection of the control triangle on the plane of reference, by means of Equations XII. v

9. Plot the triangle A, B, C, on the screen 26.

10. Project the first photograph upon the screen and secure coincidence between the projeeted image and the plotted triangle A, B, C, in the manner described. The angular adjustment of the camera remains the same as before it being only necessary to adjust the distance between the camera and screen and to adjust the angularity of the screen until the coincidence is secured.

11. Substitute a photographic plate for the screen 26, the angular adjustments of the screen holder remaining unchanged, and expose this plate so as to procure a positive reproduction of the photographic image projected thereon, this positive reproduction representing the plane-of-reference projection;

ly locate them with reference to the plotted 7 points on the map surface by procuring co1n-' cidence between the projections of the control triangles appearing in the plane-of-reference projections and the corresponding plotted points on the map surface.

15. Plot on the map surface the points W, and W appearing in the two plane-of-reference projections, these points being the contral projections on the plane of reference of the point W whose position is to be determined.

16. Draw on the maprsurface the lines P, W, and P W the intersection of which fixed the point W which is the map position of the point W.

17. Measure the distance W W, and P, W, (or W W, etc.) and from these distances and the height of the point P, (or P as the case may be), already known, calculate the height W W Theposition of W is now definitely determined and obviously the position of any similar point shown in both photographs may be determined in like manner so that the area common to photographs may be completely mapped.

A comparison ofthe hypothetical procedure of carrying out this method with the practical process described leads at once to the conclusion that there is a wide latitude in the choice of the mechanical means of dealing with the original photographs to obtain the plane-of-reference projections required for actual mapping. It seems probable, however, that apparatus constructed on the general principles of that shown in Figure 3 will be found most suitable in practice.

But if this apparatus or similar mechanical means is employed there still remain some important variations of procedure which are worth considering. Two of these will be discussed.

(1) After a coincidence has been effected between the projected triangle and the first triangle plotted on the screen, similar to the control triangle in nature in its own plane, 

